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arxiv: 2605.02883 · v1 · submitted 2026-05-04 · ✦ hep-th · cond-mat.str-el· hep-ph

Recognition: 3 theorem links

· Lean Theorem

de Sitter Vacua & pUniverses

Jeremias Aguilera-Damia , Dionysios Anninos , Tarek Anous , Johnny Gleeson , Alan Rios Fukelman
Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:12 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-ph
keywords p-Schwinger modelde Sitter spacespontaneous symmetry breakingHadamard statesde Sitter vacuadiscrete global symmetriesquantum gravity
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The pith

The p-Schwinger model on de Sitter space admits p distinct de Sitter-invariant vacua that break its discrete symmetries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the p-Schwinger model, where massless fermions carry integer charge p, on a global de Sitter background. In flat space the model spontaneously breaks discrete zero- and one-form global symmetries, producing p degenerate ground states. The authors show these features survive in de Sitter, yielding p distinct yet locally indistinguishable states that remain de Sitter invariant and obey the Hadamard condition. They further couple a multi-flavor version to quantum gravity with positive cosmological constant and obtain a semiclassical de Sitter saddle at large flavor number. The p states are interpreted as candidate microstates of the de Sitter horizon in the effective theory.

Core claim

In the p-Schwinger model on global de Sitter the discrete zero- and one-form global symmetries are spontaneously broken, producing p distinct de Sitter invariant states that are locally indistinguishable and satisfy the Hadamard property. Coupling a variant with N_f flavors to quantum gravity with positive Lambda yields a semiclassical de Sitter saddle in the large N_f limit, where the p vacua are speculatively interpreted as microstates of the de Sitter horizon in the low-energy effective field theory.

What carries the argument

The p-Schwinger model, in which charged massless fermions carry non-unit integer charge p, inducing discrete zero- and one-form global symmetries that break spontaneously into p vacua.

If this is right

  • Discrete global symmetries can be spontaneously broken in quantum field theories in de Sitter space.
  • The theory supports p locally indistinguishable de Sitter vacua that satisfy the Hadamard property.
  • A semiclassical de Sitter saddle exists in the gravitational theory at large number of flavors.
  • The p vacua supply candidate microstates for the de Sitter horizon within the low-energy effective field theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a concrete example in which the degeneracy of de Sitter vacua is tied to a discrete symmetry, offering a potential counting mechanism for horizon entropy in simple models.
  • Similar discrete-symmetry breaking patterns may appear in other de Sitter quantum field theories and could be tested by examining correlation functions across the different states.
  • If the microstate interpretation holds, it would imply that transitions between the p vacua require non-local information not accessible to local observers.

Load-bearing premise

The p states remain de Sitter invariant and Hadamard when the model is coupled to gravity, and the large-N_f limit produces a reliable semiclassical saddle without uncontrolled backreaction or higher-order corrections.

What would settle it

Explicit computation of local correlation functions in each of the p candidate vacua to confirm they are identical while the global states differ, or detection of large backreaction effects that destroy the semiclassical saddle at finite N_f.

Figures

Figures reproduced from arXiv: 2605.02883 by Alan Rios Fukelman, Dionysios Anninos, Jeremias Aguilera-Damia, Johnny Gleeson, Tarek Anous.

Figure 1
Figure 1. Figure 1: Screening of the electric field created by an view at source ↗
read the original abstract

We analyze a simple extension of the Schwinger model, which we refer to as the $p$-Schwinger model, on a de Sitter background. In this theory, the charged massless fermions carry non-unit integer charge $p$. In Minkowski space, the $p$-Schwinger model has discrete zero- and one-form global symmetries that are spontaneously broken, yielding $p$ degenerate ground states. We demonstrate that these features persist upon placing the $p$-Schwinger model on a global de Sitter background, establishing that such discrete global symmetries can be spontaneously broken for quantum field theories in de Sitter space. In particular, the theory is endowed with $p$ distinct, but locally-indistinguishable, de Sitter invariant states, the de Sitter vacua, satisfying the Hadamard property. We couple a variant of the $p$-Schwinger model with ${\rm N}_{\rm f}$ flavors to quantum gravity with $\Lambda>0$, and demonstrate the existence of a semiclassical de Sitter saddle at large ${\rm N}_{\rm f}$. In the gravitational theory, the $p$ de Sitter invariant vacua are speculatively interpreted as microstates of the de Sitter horizon in the low-energy effective field theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the p-Schwinger model (massless fermions of integer charge p) on global de Sitter space. It claims that the discrete zero- and one-form global symmetries remain spontaneously broken, producing p distinct but locally indistinguishable de Sitter-invariant states that satisfy the Hadamard condition. A multi-flavor variant with N_f flavors is then coupled to gravity with positive cosmological constant, yielding a semiclassical de Sitter saddle at large N_f; the p vacua are speculatively interpreted as microstates of the de Sitter horizon in the low-energy EFT.

Significance. If the central claims are substantiated with explicit derivations, the work would supply a concrete QFT example of spontaneous breaking of discrete symmetries in de Sitter space together with a controlled large-N_f semiclassical saddle. The emphasis on Hadamard states and the persistence of symmetry breaking on a curved background would be a useful addition to the literature on QFT in de Sitter, though the microstate interpretation remains conjectural and does not yet supply new falsifiable predictions.

major comments (2)
  1. [gravitational coupling section] Section on gravitational coupling and large-N_f limit: the existence of a reliable semiclassical de Sitter saddle is asserted without quantitative control on the backreaction of the stress-energy tensor evaluated in the p distinct vacua; no bound is given on possible O(1) deviations from the pure dS geometry induced by the discrete symmetry breaking when Lambda > 0, nor on the size of 1/N_f corrections to the metric.
  2. [p-Schwinger model on de Sitter section] Section establishing the p de Sitter vacua: the claim that the p states remain de Sitter invariant and satisfy the Hadamard property is load-bearing for the central result, yet the provided text supplies no explicit two-point function computation or verification that the states are invariant under the full de Sitter isometry group while remaining locally indistinguishable.
minor comments (2)
  1. [title and abstract] The title contains 'pUniverses' but the abstract and text do not define or motivate this terminology; a brief clarification of its relation to the multiple vacua would improve readability.
  2. [notation] Notation for the number of flavors alternates between N_f and {rm N}_{rm f}; standardize throughout for consistency.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and have revised the manuscript accordingly to provide the requested quantitative estimates and explicit verifications.

read point-by-point responses

  1. Referee: Section on gravitational coupling and large-N_f limit: the existence of a reliable semiclassical de Sitter saddle is asserted without quantitative control on the backreaction of the stress-energy tensor evaluated in the p distinct vacua; no bound is given on possible O(1) deviations from the pure dS geometry induced by the discrete symmetry breaking when Lambda > 0, nor on the size of 1/N_f corrections to the metric.

    Authors: We agree that a quantitative bound on backreaction strengthens the claim of a reliable semiclassical saddle. In the large-N_f limit the matter contribution to the stress-energy tensor is O(N_f) before normalization by the overall action scaling, so that its effect on the metric is suppressed by 1/N_f. We have added a new paragraph in the gravitational-coupling section that estimates the size of the deviation from pure de Sitter geometry and shows that the O(1/N_f) corrections remain perturbatively small for N_f sufficiently large compared to the ratio of the de Sitter radius to the ultraviolet cutoff. This establishes that the saddle stays close to the pure de Sitter background. revision: yes

  2. Referee: Section establishing the p de Sitter vacua: the claim that the p states remain de Sitter invariant and satisfy the Hadamard property is load-bearing for the central result, yet the provided text supplies no explicit two-point function computation or verification that the states are invariant under the full de Sitter isometry group while remaining locally indistinguishable.

    Authors: The p vacua are defined as the unique de Sitter-invariant states in each of the p superselection sectors associated with the spontaneously broken discrete symmetries. Because the short-distance singularities of the two-point functions are fixed by the local Minkowski structure, the Hadamard condition follows automatically and is independent of the global de Sitter geometry. We acknowledge that an explicit verification was not included in the original text. We have added an appendix that writes the two-point functions in global de Sitter coordinates, demonstrates invariance under the full de Sitter isometry group, and confirms that the states remain locally indistinguishable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces the p-Schwinger model, analyzes its discrete symmetries and vacua first in Minkowski space then on global de Sitter, and performs a large-N_f semiclassical analysis when coupled to gravity with positive Lambda. These steps rely on direct model construction and standard large-N saddle-point techniques rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation that reduces the central claims to tautology. The microstate interpretation is explicitly labeled speculative and is not used to derive any result. No equations or sections exhibit the patterns of self-definitional closure or imported uniqueness from prior author work that would force the outcome by construction. The analysis therefore stands as an independent examination of the model on de Sitter.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claims rest on the known spontaneous symmetry breaking in the flat-space p-Schwinger model and the validity of the large-N_f semiclassical limit when gravity is included.

free parameters (2)
  • p
    Integer charge of the fermions; determines the number of degenerate vacua.
  • N_f
    Number of fermion flavors; taken large to obtain the semiclassical de Sitter saddle.
axioms (1)
  • domain assumption The p-Schwinger model in Minkowski space has discrete zero- and one-form global symmetries that are spontaneously broken, yielding p degenerate ground states.
    Invoked as established prior knowledge to extend the model to de Sitter.
invented entities (1)
  • p de Sitter vacua interpreted as microstates of the de Sitter horizon no independent evidence
    purpose: To connect the QFT vacua to quantum gravity descriptions of the horizon.
    This is a speculative interpretation without independent evidence or falsifiable predictions supplied.

pith-pipeline@v0.9.0 · 5542 in / 1572 out tokens · 81215 ms · 2026-05-08T18:12:30.512155+00:00 · methodology

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Reference graph

Works this paper leans on

81 extracted references · 46 canonical work pages · 6 internal anchors

  1. [1]

    Schwinger Model on S(2),

    C. Jayewardena, “Schwinger Model on S(2),”Helv. Phys. Acta61(1988) 636–711

    1988

  2. [2]

    Anninos, T

    D. Anninos, T. Anous, and A. Rios Fukelman, “De Sitter at all loops: the story of the Schwinger model,”JHEP08(2024) 155,arXiv:2403.16166 [hep-th]

    work page arXiv 2024

  3. [3]

    A Note on the Perturbative Expansion of the Schwinger Model onS 2,

    J. Smith, “A Note on the Perturbative Expansion of the Schwinger Model onS 2,” arXiv:2603.21938 [hep-th]

    work page arXiv

  4. [4]

    Gauge Invariance and Mass. 2.,

    J. S. Schwinger, “Gauge Invariance and Mass. 2.,”Phys. Rev.128(1962) 2425–2429

    1962

  5. [5]

    Fractionalθangle, ’t Hooft anomaly, and quantum instantons in charge-qmulti-flavor Schwinger model,

    T. Misumi, Y. Tanizaki, and M. ¨Unsal, “Fractionalθangle, ’t Hooft anomaly, and quantum instantons in charge-qmulti-flavor Schwinger model,”JHEP07(2019) 018, arXiv:1905.05781 [hep-th]

    work page arXiv 2019

  6. [6]

    Instantons, the Quark Model, and the 1/n Expansion,

    E. Witten, “Instantons, the Quark Model, and the 1/n Expansion,”Nucl. Phys. B149(1979) 285–320. 53

    1979

  7. [7]

    Symmetries and strings of adjoint QCD2,

    Z. Komargodski, K. Ohmori, K. Roumpedakis, and S. Seifnashri, “Symmetries and strings of adjoint QCD2,”JHEP03(2021) 103,arXiv:2008.07567 [hep-th]

    work page arXiv 2021

  8. [8]

    Vacuum structure of charge k two-dimensional QED and dynamics of an anti D-string near an O1 −-plane,

    A. Armoni and S. Sugimoto, “Vacuum structure of charge k two-dimensional QED and dynamics of an anti D-string near an O1 −-plane,”JHEP03(2019) 175,arXiv:1812.10064 [hep-th]

    work page arXiv 2019

  9. [9]

    Modifying the Sum Over Topological Sector s and Constraints on Super- gravity,

    N. Seiberg, “Modifying the Sum Over Topological Sectors and Constraints on Supergravity,” JHEP07(2010) 070,arXiv:1005.0002 [hep-th]

    work page arXiv 2010

  10. [10]

    Iqbal,Lectures on Generalized Global Symmetries

    N. Iqbal,Lectures on Generalized Global Symmetries. SpringerBriefs in Physics. Springer Cham, 8, 2025

    2025

  11. [11]

    Canonical bf type topological field theory and fractional statistics of strings,

    M. Bergeron, G. W. Semenoff, and R. J. Szabo, “Canonical bf type topological field theory and fractional statistics of strings,”Nucl. Phys. B437(1995) 695–722, arXiv:hep-th/9407020

    work page arXiv 1995

  12. [12]

    Lectures on 2-d gauge theories: Topological aspects and path integral techniques,

    M. Blau and G. Thompson, “Lectures on 2-d gauge theories: Topological aspects and path integral techniques,” inSummer School in High-energy Physics and Cosmology (Includes Workshop on Strings, Gravity, and Related Topics 29-30 Jul 1993), pp. 0175–244. 10, 1993. arXiv:hep-th/9310144

    work page internal anchor Pith review arXiv 1993

  13. [13]

    Coupling a QFT to a TQFT and Duality

    A. Kapustin and N. Seiberg, “Coupling a QFT to a TQFT and Duality,”JHEP04(2014) 001,arXiv:1401.0740 [hep-th]

    work page Pith review arXiv 2014

  14. [14]

    PiTP 2015 – “Fun with Free Field Theory

    I. for Advanced Study, “PiTP 2015 – “Fun with Free Field Theory” – Nathan Seiberg,” July 27, 2015.https://youtu.be/pqgNrVTQ4yM. YouTube video

    2015

  15. [15]

    Symmetries, universes and phases of QCD 2 with an adjoint Dirac fermion,

    J. A. Damia, G. Galati, and L. Tizzano, “Symmetries, universes and phases of QCD 2 with an adjoint Dirac fermion,”JHEP12(2025) 230,arXiv:2409.17989 [hep-th]

    work page arXiv 2025

  16. [16]

    Particle Creation in de Sitter Space,

    E. Mottola, “Particle Creation in de Sitter Space,”Phys. Rev. D31(1985) 754

    1985

  17. [17]

    Vacuum States in de Sitter Space,

    B. Allen, “Vacuum States in de Sitter Space,”Phys. Rev. D32(1985) 3136

    1985

  18. [18]

    Conformal vacua and entropy in de Sitter space

    R. Bousso, A. Maloney, and A. Strominger, “Conformal vacua and entropy in de Sitter space,”Phys. Rev. D65(2002) 104039,arXiv:hep-th/0112218

    work page arXiv 2002

  19. [19]

    Path integral games with de Sitterα-vacua,

    N. Miller, “Path integral games with de Sitterα-vacua,”JHEP10(2025) 097, arXiv:2503.13701 [hep-th]

    work page arXiv 2025

  20. [20]

    Lifetimes of near eternal false vacua,

    A. Cherman and T. Jacobson, “Lifetimes of near eternal false vacua,”Phys. Rev. D103 no. 10, (2021) 105012,arXiv:2012.10555 [hep-th]

    work page arXiv 2021

  21. [21]

    Wave Function of the Universe,

    J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,”Phys. Rev. D28(1983) 2960–2975

    1983

  22. [22]

    Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors,

    I. R. Klebanov, S. S. Pufu, S. Sachdev, and B. R. Safdi, “Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors,”JHEP05(2012) 036,arXiv:1112.5342 [hep-th]

    work page arXiv 2012

  23. [23]

    Klebanov and Grigory Tarnopolsky,Conformal QEDd,F-Theorem and theϵExpansion,J

    S. Giombi, I. R. Klebanov, and G. Tarnopolsky, “Conformal QED d,F-Theorem and theϵ Expansion,”J. Phys. A49no. 13, (2016) 135403,arXiv:1508.06354 [hep-th]

    work page arXiv 2016

  24. [24]

    Kitaev and J

    A. Kitaev and J. Preskill, “Topological entanglement entropy,”Phys. Rev. Lett.96(2006) 110404,arXiv:hep-th/0510092

    work page arXiv 2006

  25. [25]

    Detecting Topological Order in a Ground State Wave Function,

    M. Levin and X.-G. Wen, “Detecting Topological Order in a Ground State Wave Function,” Phys. Rev. Lett.96(2006) 110405,arXiv:cond-mat/0510613. 54

    work page arXiv 2006

  26. [26]

    Gauge Theories of Vector Particles,

    J. Schwinger, A. P. Balachandran, B. Jaksic, I. Saavedra, J. Kvasnica, P. W. D. Nielsen, C. Boulware, P. Bollini, and Rastall, “Gauge Theories of Vector Particles,” inTheoretical Physics, pp. 89–134. IAEA, Vienna, 1963

    1963

  27. [27]

    Quantum electrodynamics in two-dimensions,

    J. H. Lowenstein and J. A. Swieca, “Quantum electrodynamics in two-dimensions,”Annals Phys.68(1971) 172–195

    1971

  28. [28]

    Vector Meson Mass Generation Through Chiral Anomalies,

    R. Jackiw and R. Rajaraman, “Vector Meson Mass Generation Through Chiral Anomalies,” Phys. Rev. Lett.54(1985) 1219. [Erratum: Phys.Rev.Lett. 54, 2060 (1985)]

    1985

  29. [29]

    Charge Shielding and Quark Confinement in the Massive Schwinger Model,

    S. R. Coleman, R. Jackiw, and L. Susskind, “Charge Shielding and Quark Confinement in the Massive Schwinger Model,”Annals Phys.93(1975) 267

    1975

  30. [30]

    More About the Massive Schwinger Model,

    S. R. Coleman, “More About the Massive Schwinger Model,”Annals Phys.101(1976) 239

    1976

  31. [31]

    Comment on Fujikawa’s Analysis Applied to the Schwinger Model,

    R. Roskies and F. Schaposnik, “Comment on Fujikawa’s Analysis Applied to the Schwinger Model,”Phys. Rev. D23(1981) 558–560

    1981

  32. [32]

    THE SCHWINGER MODEL IN CURVED SPACE-TIME,

    R. Gass, “THE SCHWINGER MODEL IN CURVED SPACE-TIME,”Phys. Rev. D27 (1983) 2893–2905

    1983

  33. [33]

    Thermodynamics of the Schwinger model in a two-dimensional de Sitter space-time,

    T. Oki, Y. Osada, and Y. Tanikawa, “Thermodynamics of the Schwinger model in a two-dimensional de Sitter space-time,”Bull. Okayama Univ. Sci.A20(1984) 97–108

    1984

  34. [34]

    Chiral Schwinger Model in Curved Space-time,

    J. Barcelos-Neto and A. K. Das, “Chiral Schwinger Model in Curved Space-time,”Z. Phys. C 32(1986) 527

    1986

  35. [35]

    Field theories on the Poincare disk,

    F. Ferrari, “Field theories on the Poincare disk,”Int. J. Mod. Phys. A11(1996) 5389–5404, arXiv:hep-th/9502104

    work page arXiv 1996

  36. [36]

    Path Integral Measure for Gauge Invariant Fermion Theories,

    K. Fujikawa, “Path Integral Measure for Gauge Invariant Fermion Theories,”Phys. Rev. Lett. 42(1979) 1195–1198

    1979

  37. [37]

    Generalized Global Symmetries

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP02(2015) 172,arXiv:1412.5148 [hep-th]

    work page internal anchor Pith review arXiv 2015

  38. [38]

    The Quantum Sine-Gordon Equation as the Massive Thirring Model,

    S. R. Coleman, “The Quantum Sine-Gordon Equation as the Massive Thirring Model,”Phys. Rev. D11(1975) 2088

    1975

  39. [39]

    Soliton Operators for the Quantized Sine-Gordon Equation,

    S. Mandelstam, “Soliton Operators for the Quantized Sine-Gordon Equation,”Phys. Rev. D 11(1975) 3026

    1975

  40. [40]

    Bosonization and QCD in two-dimensions,

    Y. Frishman and J. Sonnenschein, “Bosonization and QCD in two-dimensions,”Phys. Rept. 223(1993) 309–348,arXiv:hep-th/9207017

    work page arXiv 1993

  41. [41]

    Ji, S.-H

    W. Ji, S.-H. Shao, and X.-G. Wen, “Topological Transition on the Conformal Manifold,” Phys. Rev. Res.2no. 3, (2020) 033317,arXiv:1909.01425 [cond-mat.str-el]

    work page arXiv 2020

  42. [42]

    ‘The discreet charm of the discrete series in dS2’.J

    D. Anninos, T. Anous, B. Pethybridge, and G. S ¸eng¨ or, “The Discreet Charm of the Discrete Series in DS 2,”arXiv:2307.15832 [hep-th]

    work page arXiv

  43. [43]

    Finite temperature Schwinger model,

    I. Sachs and A. Wipf, “Finite temperature Schwinger model,”Helv. Phys. Acta65(1992) 652–678,arXiv:1005.1822 [hep-th]

    work page arXiv 1992

  44. [44]

    Topological Fluctuations and Breaking of Chiral Symmetry in Gauge Theories Involving Massless Fermions,

    N. K. Nielsen and B. Schroer, “Topological Fluctuations and Breaking of Chiral Symmetry in Gauge Theories Involving Massless Fermions,”Nucl. Phys. B120(1977) 62–76

    1977

  45. [45]

    Path Integral Representations for Tunneling Amplitudes in the Schwinger Model,

    K. D. Rothe and J. A. Swieca, “Path Integral Representations for Tunneling Amplitudes in the Schwinger Model,”Annals Phys.117(1979) 382

    1979

  46. [46]

    Four-fermion 55 deformations of the massless Schwinger model and confinement,

    A. Cherman, T. Jacobson, M. Shifman, M. Unsal, and A. Vainshtein, “Four-fermion 55 deformations of the massless Schwinger model and confinement,”JHEP01(2023) 087, arXiv:2203.13156 [hep-th]

    work page arXiv 2023

  47. [47]

    RG flows in 2d QCD,

    D. Delmastro and J. Gomis, “RG flows in 2d QCD,”JHEP09(2023) 158, arXiv:2211.09036 [hep-th]

    work page arXiv 2023

  48. [48]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, and D. Senechal,Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997

    1997

  49. [49]

    Nonabelian Bosonization and Multiflavor QED and QCD in Two-dimensions,

    D. Gepner, “Nonabelian Bosonization and Multiflavor QED and QCD in Two-dimensions,” Nucl. Phys. B252(1985) 481–507

    1985

  50. [50]

    On the Realization of Chiral Symmetry in (1+1)-dimensions,

    I. Affleck, “On the Realization of Chiral Symmetry in (1+1)-dimensions,”Nucl. Phys. B265 (1986) 448–468

    1986

  51. [51]

    Dempsey, I

    R. Dempsey, I. R. Klebanov, S. S. Pufu, B. T. Søgaard, and B. Zan, “Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature,”Phys. Rev. Lett.132no. 3, (2024) 031603,arXiv:2305.04437 [hep-th]

    work page arXiv 2024

  52. [52]

    Infrared phases of 2d QCD,

    D. Delmastro, J. Gomis, and M. Yu, “Infrared phases of 2d QCD,”JHEP02(2023) 157, arXiv:2108.02202 [hep-th]

    work page arXiv 2023

  53. [53]

    Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls

    M. M. Anber and E. Poppitz, “Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls,”JHEP09(2018) 076,arXiv:1807.00093 [hep-th]

    work page arXiv 2018

  54. [54]

    Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj),

    M. M. Anber and E. Poppitz, “Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj),”JHEP05(2019) 151,arXiv:1811.10642 [hep-th]

    work page arXiv 2019

  55. [55]

    Consequences of anomalous Ward identities,

    J. Wess and B. Zumino, “Consequences of anomalous Ward identities,”Phys. Lett. B37 (1971) 95–97

    1971

  56. [56]

    Global Aspects of Current Algebra,

    E. Witten, “Global Aspects of Current Algebra,”Nucl. Phys. B223(1983) 422–432

    1983

  57. [57]

    Quantum Gravity in Two-Dimensions,

    A. M. Polyakov, “Quantum Gravity in Two-Dimensions,”Mod. Phys. Lett. A2(1987) 893

    1987

  58. [58]

    Anninos, C

    D. Anninos, C. Baracco, and B. M¨ uhlmann, “Remarks on 2D quantum cosmology,”JCAP10 (2024) 031,arXiv:2406.15271 [hep-th]

    work page arXiv 2024

  59. [59]

    Anninos, T

    D. Anninos, T. Bautista, and B. M¨ uhlmann, “The two-sphere partition function in two-dimensional quantum gravity,”JHEP09(2021) 116,arXiv:2106.01665 [hep-th]

    work page arXiv 2021

  60. [60]

    The two-sphere partition function in two-dimensional quantum gravity at fixed area,

    B. M¨ uhlmann, “The two-sphere partition function in two-dimensional quantum gravity at fixed area,”JHEP09no. 189, (2021) 189,arXiv:2106.04532 [hep-th]

    work page arXiv 2021

  61. [61]

    Conformal Field Theory and 2D Quantum Gravity,

    J. Distler and H. Kawai, “Conformal Field Theory and 2D Quantum Gravity,”Nucl. Phys. B 321(1989) 509–527

    1989

  62. [62]

    Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge,

    F. David, “Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge,”Mod. Phys. Lett. A3(1988) 1651

    1988

  63. [63]

    Ginsparg and G

    P. H. Ginsparg and G. W. Moore, “Lectures on 2-D gravity and 2-D string theory,” in Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, pp. 277–469. 10, 1993.arXiv:hep-th/9304011

    work page internal anchor Pith review arXiv 1993

  64. [64]

    Notes on matrix models (matrix musings),

    D. Anninos and B. M¨ uhlmann, “Notes on matrix models (matrix musings),”J. Stat. Mech. 2008(2020) 083109,arXiv:2004.01171 [hep-th]

    work page arXiv 2008

  65. [65]

    Path Integrals and the Indefiniteness of the Gravitational Action,

    G. W. Gibbons, S. W. Hawking, and M. J. Perry, “Path Integrals and the Indefiniteness of the Gravitational Action,”Nucl. Phys. B138(1978) 141–150

    1978

  66. [66]

    Rigorous results for timelike Liouville field theory

    S. Chatterjee, “Rigorous results for timelike Liouville field theory,”arXiv:2504.02348 [math.PR]. 56

    work page internal anchor Pith review Pith/arXiv arXiv

  67. [67]

    Allameh and E

    K. Allameh and E. Shaghoulian, “Timelike Liouville theory and AdS 3 gravity at finite cutoff,”arXiv:2508.03236 [hep-th]

    work page arXiv

  68. [68]

    BRST cohomology of timelike Liouville theory,

    T. Bautista, H. Erbin, and M. Kudrna, “BRST cohomology of timelike Liouville theory,” JHEP05(2020) 029,arXiv:2002.01722 [hep-th]

    work page arXiv 2020

  69. [69]

    Symmetries and Strings in Field Theory and Gravity

    T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and Gravity,”Phys. Rev. D83(2011) 084019,arXiv:1011.5120 [hep-th]

    work page Pith review arXiv 2011

  70. [70]

    Monopoles, Duality, and String Theory

    J. Polchinski, “Monopoles, duality, and string theory,”Int. J. Mod. Phys. A19S1(2004) 145–156,arXiv:hep-th/0304042

    work page Pith review arXiv 2004

  71. [71]

    Three-dimensional de Sitter horizon thermodynamics,

    D. Anninos and E. Harris, “Three-dimensional de Sitter horizon thermodynamics,”JHEP10 (2021) 091,arXiv:2106.13832 [hep-th]

    work page arXiv 2021

  72. [72]

    De Sitter Horizon Edge Partition Functions

    Y. T. A. Law, “De Sitter Horizon Edge Partition Functions,”arXiv:2501.17912 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  73. [73]

    Horizon Edge Partition Functions in $\Lambda>0$ Quantum Gravity

    Y. T. A. Law and V. Lochab, “Horizon Edge Partition Functions in Λ>0 Quantum Gravity,”arXiv:2603.20913 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  74. [74]

    Fate of stringy noninvertible symmetries,

    J. J. Heckman, J. McNamara, M. Montero, A. Sharon, C. Vafa, and I. Valenzuela, “Fate of stringy noninvertible symmetries,”Phys. Rev. D110no. 10, (2024) 106001, arXiv:2402.00118 [hep-th]

    work page arXiv 2024

  75. [75]

    Black holes as red herrings: Topological fluctuations and the loss of quantum coherence,

    S. R. Coleman, “Black holes as red herrings: Topological fluctuations and the loss of quantum coherence,”Nucl. Phys. B307(1988) 867–882

    1988

  76. [76]

    Wormholes and the Cosmological Constant,

    I. R. Klebanov, L. Susskind, and T. Banks, “Wormholes and the Cosmological Constant,” Nucl. Phys. B317(1989) 665–692

    1989

  77. [77]

    The Alpha parameters of wormholes,

    S. W. Hawking, “The Alpha parameters of wormholes,”Phys. Scripta T36(1991) 222–227

    1991

  78. [78]

    Heat kernel expansion: user’s manual,

    D. Vassilevich, “Heat kernel expansion: user’s manual,”Physics Reports388no. 5, (2003) 279–360.https://www.sciencedirect.com/science/article/pii/S0370157303003545

    2003

  79. [79]

    Anninos, F

    D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, “Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions,”JHEP01(2022) 088,arXiv:2009.12464 [hep-th]

    work page arXiv 2022

  80. [80]

    Lectures on generalized symmetries,

    L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre, and H. Tillim, “Lectures on generalized symmetries,”Phys. Rept.1051(2024) 1–87, arXiv:2307.07547 [hep-th]

    work page arXiv 2024

Showing first 80 references.